The compactness of \(\operatorname{\mathcal {O}_{ad}}\)  follows from the fact that it is closed and relatively compact – as defined by Chenais, D. [1]} – with respect to the topology on \(\mathcal {U}_{ad}\) . One can also read upon the proof in [2]}. We shall not discuss the proof of such properties, nevertheless they are summarized on the lemma below.
